Fault-tolerant diameter for three family interconnection networks

Let G=(V,E) be an k-connected graph. The (k?1)-fault-tolerant diameter of G, denoted by D _k (G), is defined as D _k (G)=max?{d(G?F)|F?V(G)?and?|F|=k?1}, where G?F denotes the subgraph induced by V(G)?F. The fault-tolerant diameter is one important parameter for measuring the reliability and efficiency of interconnection networks. In this paper, we will give the bounds of fault-tolerant diameter of three family interconnection networks.     

The total {k}-domatic number of wheels and complete graphs

Let k be a positive integer and let G be a graph with vertex set V(G). The total {k}-dominating function (T{k}DF) of a graph G is a function f from V(G) to the set {0,1,2,??,k}, such that for each vertex v??V(G), the sum of the values of all its neighbors assigned by f is at least k. A set {f ₁,f ₂,??,f _d } of pairwise different T{k}DFs of G with the property that $\sum_{i=1}^{d}f_{i}(v)\leq k$ for each v??V(G), is called a total {k}-dominating family (T{k}D family) of G. The total {k}-domatic number of a graph G, denoted by $d_{t}^{\{k\}}(G)$ , is the maximum number of functions in a T{k}D family. In this paper, we determine the exact values of the total {k}-domatic numbers of wheels and complete graphs, which answers an open problem of Sheikholeslami and Volkmann (J. Comb. Optim., 2010) and completes a result in the same paper.     

Note on the hardness of generalized connectivity

Let G be a nontrivial connected graph of order n and let k be an integer with 2??k??n. For a set S of k vertices of G, let ??(S) denote the maximum number ? of edge-disjoint trees T ₁,T ₂,??,T _? in G such that V(T _i )??V(T _j )=S for every pair i,j of distinct integers with 1??i,j???. Chartrand et al. generalized the concept of connectivity as follows: The k-connectivity, denoted by ?? _k (G), of G is defined by ?? _k (G)=min{??(S)}, where the minimum is taken over all k-subsets S of V(G). Thus ?? ₂(G)=??(G), where ??(G) is the connectivity of G, for which there are polynomial-time algorithms to solve it. This paper mainly focus on the complexity of determining the generalized connectivity of a graph. At first, we obtain that for two fixed positive integers k ₁ and k ₂, given a graph G and a k ₁-subset S of V(G), the problem of deciding whether G contains k ₂ internally disjoint trees connecting S can be solved by a polynomial-time algorithm. Then, we show that when k ₁ is a fixed integer of at least 4, but k ₂ is not a fixed integer, the problem turns out to be NP-complete. On the other hand, when k ₂ is a fixed integer of at least 2, but k ₁ is not a fixed integer, we show that the problem also becomes NP-complete.     

L(d,1)-labelings of the edge-path-replacement of a graph

For two positive integers j and k with j≥k, an L(j,k)-labeling of a graph G is an assignment of nonnegative integers to V(G) such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L(j,k)-labeling of a graph G is the difference between the maximum and minimum integers used by it. The L(j,k)-labelings-number of G is the minimum span over all L(j,k)-labelings of G. This paper focuses on L(d,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G. Note that G(P ₃) is the incidence graph of G. L(d,1)-labelings of the edge-path-replacement G(P _k ) of a graph, called (d,1)-total labeling of G, was introduced in 2002 by Havet and Yu (Workshop graphs and algorithms, 2003; Discrete Math 308:493–513, 2008). Havet and Yu (Discrete Math 308:498–513, 2008) obtained the bound $\Delta+ d-1\leq\lambda^{T}_{d}(G)\leq2\Delta+ d-1$ and conjectured $\lambda^{T}_{d}(G)\leq\Delta+2d-1$ . In (Lü in J Comb Optim, to appear; Zhejiang University, submitted), we worked on L(2,1)-labelings-number and L(1,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G, and obtained that λ(G(P _k ))≤Δ+2 for k≥5, and conjecture λ(G(P ₄))≤Δ+2 for any graph G with maximum degree Δ. In this paper, we will study L(d,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G for d≥3 and k≥4.     

The total {k}-domatic number of a graph

For a positive integer k, a total {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0,1,2,…,k} such that for any vertex v∈V(G), the condition ∑ _u∈N(v) f(u)≥k is fulfilled, where N(v) is the open neighborhood of v. A set {f ₁,f ₂,…,f _d } of total {k}-dominating functions on G with the property that ?_i=1^df_i(v) ￡ k\sum_{i=1}^{d}f_{i}(v)\le k for each v∈V(G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by d_t^{k}(G)d_{t}^{\{k\}}(G). Note that d_t^{1}(G)d_{t}^{\{1\}}(G) is the classic total domatic number d _t (G). In this paper we initiate the study of the total {k}-domatic number in graphs and we present some bounds for d_t^{k}(G)d_{t}^{\{k\}}(G). Many of the known bounds of d _t (G) are immediate consequences of our results.     

Max-coloring paths: tight bounds and extensions

The max-coloring problem is to compute a legal coloring of the vertices of a graph G=(V,E) with vertex weights w such that $\sum_{i=1}^{k}\max_{v\in C_{i}}w(v_{i})$ is minimized, where C ₁,??,C _k are the various color classes. For general graphs, max-coloring is as hard as the classical vertex coloring problem, a special case of the former where vertices have unit weight. In fact, in some cases it can even be harder: for example, no polynomial time algorithm is known for max-coloring trees. In this paper we consider the problem of max-coloring paths and its generalization, max-coloring skinny trees, a broad class of trees that includes paths and spiders. For these graphs, we show that max-coloring can be solved in time O(|V|+time for sorting the vertex weights). When vertex weights are real numbers, we show a matching lower bound of ??(|V|log?|V|) in the algebraic computation tree model.     

L(2,1)-labelings of the edge-path-replacement of a graph

For two positive integers j and k with j≥k, an L(j,k)-labeling of a graph G is an assignment of nonnegative integers to V(G) such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L(j,k)-labeling of a graph G is the difference between the maximum and minimum integers used by it. The L(j,k)-labelings-number of G is the minimum span over all L(j,k)-labelings of G. This paper focuses on L(2,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G. Note that G(P ₃) is the incidence graph of G. L(2,1)-labelings of the edge-path-replacement G(P ₃) of a graph, called (2,1)-total labeling of G, was introduced by Havet and Yu in 2002 (Workshop graphs and algorithms, Dijon, France, 2003; Discrete Math. 308:498–513, 2008). They (Havet and Yu, Discrete Math. 308:498–513, 2008) obtain the bound $\Delta+1\leq\lambda^{T}_{2}(G)\leq2\Delta+1$ and conjectured $\lambda^{T}_{2}(G)\leq\Delta+3$ . In this paper, we obtain that λ(G(P _k ))≤Δ+2 for k≥5, and conjecture λ(G(P ₄))≤Δ+2 for any graph G with maximum degree Δ.     

Generalized perfect domination in graphs

Let k be a positive integer and G=(V,E) be a graph. A vertex subset D of a graph G is called a perfect k-dominating set of G, if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γ _kp (G). In this paper, we give characterizations of graphs for which γ _kp (G)=γ(G)+k?2 and prove that the perfect k-domination problem is NP-complete even when restricted to bipartite graphs and chordal graphs. Also, by using dynamic programming techniques, we obtain an algorithm to determine the perfect k-domination number of trees.     

Some results on the injective chromatic number of?graphs

A k-coloring of a graph G=(V,E) is a mapping c:V??{1,2,??,k}. The coloring c is injective if, for every vertex v??V, all the neighbors of v are assigned with distinct colors. The injective chromatic number ?? _i (G) of G is the smallest k such that G has an injective k-coloring. In this paper, we prove that every K ₄-minor free graph G with maximum degree ????1 has $\chi_{i}(G)\le \lceil \frac{3}{2}\Delta\rceil$ . Moreover, some related results and open problems are given.     

On L(2,1)-labeling of generalized Petersen graphs

A variation of the classical channel assignment problem is to assign a radio channel which is a nonnegative integer to each radio transmitter so that ??close?? transmitters must receive different channels and ??very close?? transmitters must receive channels that are at least two channels apart. The goal is to minimize the span of a feasible assignment. This channel assignment problem can be modeled with distance-dependent graph labelings. A k-L(2,1)-labeling of a graph G is a mapping f from the vertex set of G to the set {0,1,2,??,k} such that |f(x)?f(y)|??2 if d(x,y)=1 and $f(x)\not =f(y)$ if d(x,y)=2, where d(x,y) is the distance between vertices x and y in G. The minimum k for which G admits an k-L(2,1)-labeling, denoted by ??(G), is called the ??-number of G. Very little is known about ??-numbers of 3-regular graphs. In this paper we focus on an important subclass of 3-regular graphs called generalized Petersen graphs. For an integer n??3, a graph G is called a generalized Petersen graph of order n if and only if G is a 3-regular graph consisting of two disjoint cycles (called inner and outer cycles) of length n, where each vertex of the outer (resp. inner) cycle is adjacent to exactly one vertex of the inner (resp. outer) cycle. In 2002, Georges and Mauro conjectured that ??(G)??7 for all generalized Petersen graphs G of order n??7. Later, Adams, Cass and Troxell proved that Georges and Mauro??s conjecture is true for orders 7 and 8. In this paper it is shown that Georges and Mauro??s conjecture is true for generalized Petersen graphs of orders 9, 10, 11 and 12.     

FPT algorithms for Connected Feedback Vertex Set

We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G=(V,E) and an integer k, decide whether there exists F?V, |F|??k, such that G[V?F] is a forest and G[F] is connected. We show that Connected Feedback Vertex Set can be solved in time O(2 ^O(k) n ^O(1)) on general graphs and in time $O(2^{O(\sqrt{k}\log k)}n^{O(1)})$ on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses, as a subroutine, a parameterized algorithm for Group Steiner Tree, a well studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree of independent interest and believe that it could be useful for obtaining parameterized algorithms for other connectivity problems.     

k-tuple total domination in cross products of graphs

For k??1 an integer, a set S of vertices in a graph G with minimum degree at least?k is a k-tuple total dominating set of G if every vertex of G is adjacent to at least k vertices in S. The minimum cardinality of a k-tuple total dominating set of G is the k-tuple total domination number of G. When k=1, the k-tuple total domination number is the well-studied total domination number. In this paper, we establish upper and lower bounds on the k-tuple total domination number of the cross product graph G×H for any two graphs G and H with minimum degree at least?k. In particular, we determine the exact value of the k-tuple total domination number of the cross product of two complete graphs.     

Incremental Facility Location Problem and Its Competitive Algorithms

We consider the incremental version of the k-Facility Location Problem, which is a common generalization of the facility location and the k-median problems. The objective is to produce an incremental sequence of facility sets F ₁?F ₂?????F _n , where each F _k contains at most k facilities. An incremental facility sequence or an algorithm producing such a sequence is called c -competitive if the cost of each F _k is at most c times the optimum cost of corresponding k-facility location problem, where c is called competitive ratio. In this paper we present two competitive algorithms for this problem. The first algorithm produces competitive ratio 8α, where α is the approximation ratio of k-facility location problem. By recently result (Zhang, Theor. Comput. Sci. 384:126–135, 2007), we obtain the competitive ratio \(16+8\sqrt{3}+\epsilon\). The second algorithm has the competitive ratio Δ+1, where Δ is the ratio between the maximum and minimum nonzero interpoint distances. The latter result has its self interest, specially for the small metric space with Δ≤8α?1.     

Circular L(j,k)-labeling number of direct product of path and cycle

Let j, k and m be positive numbers, a circular m-L(j,k)-labeling of a graph G is a function f:V(G)→[0,m) such that |f(u)?f(v)| _m ≥j if u and v are adjacent, and |f(u)?f(v)| _m ≥k if u and v are at distance two, where |a?b| _m =min{|a?b|,m?|a?b|}. The minimum m such that there exist a circular m-L(j,k)-labeling of G is called the circular L(j,k)-labeling number of G and is denoted by σ _j,k (G). In this paper, for any two positive numbers j and k with j≤k, we give some results about the circular L(j,k)-labeling number of direct product of path and cycle.     

Distance two edge labelings of lattices

Suppose G is a graph. Two edges e and e′ in G are said to be adjacent if they share a common end vertex, and distance two apart if they are nonadjacent but both are adjacent to a common edge. Let j and k be two positive integers. An L(j,k)-edge-labeling of a graph G is an assignment of nonnegative integers, called labels, to the edges of G such that the difference between labels of any two adjacent edges is at least j, and the difference between labels of any two edges that are distance two apart is at least k. The minimum range of labels over all L(j,k)-edge-labelings of a graph G is called the L(j,k)-edge-labeling number of G, denoted by $\lambda_{j,k}'(G)$ . Let m, j and k be positive integers. An m-circular-L(j,k)-edge-labeling of a graph G is an assignment f from {0,1,…,m?1} to the edges of G such that, for any two edges e and e′, |f(e)?f(e′)| _m ≥j if e and e′ are adjacent, and |f(e)?f(e′)| _m ≥k if e and e′ are distance two apart, where |a| _m =min{a,m?a}. The minimum m such that G has an m-circular-L(j,k)-edge-labeling is called the circular-L(j,k)-edge-labeling number of G, denoted by $\sigma_{j,k}'(G)$ . This paper investigates the L(1,1)-edge-labeling numbers, the L(2,1)-edge-labeling numbers and the circular-L(2,1)-edge-labeling numbers of the hexagonal lattice, the square lattice, the triangular lattice and the strong product of two infinite paths.     

Exact and approximation algorithms for?the?complementary maximal strip recovery problem

Given two genomic maps G ₁ and G ₂ each represented as a sequence of n gene markers, the maximal strip recovery (MSR) problem is to retain the maximum number of markers in both G ₁ and G ₂ such that the resultant subsequences, denoted as $G_{1}^{*}$ and $G_{2}^{*}$ , can be partitioned into the same set of maximal substrings of length greater than or equal to two. Such substrings can occur in the reversal and negated form. The complementary maximal strip recovery (CMSR) problem is to delete the minimum number of markers from both G ₁ and G ₂ for the same purpose, with its optimization goal exactly complementary to maximizing the total number of gene markers retained in the final maximal substrings. Both MSR and CMSR have been shown NP-hard and APX-hard. A?4-approximation algorithm is known for the MSR problem, but no constant ratio approximation algorithm for CMSR. In this paper, we present an O(3 ^k n ²)-time fixed-parameter tractable (FPT) algorithm, where k is the size of the optimal solution, and a 3-approximation algorithm for the CMSR problem.     

The paths embedding of the arrangement graphs with prescribed vertices in given position

Let n and k be positive integers with n?k≥2. The arrangement graph A _n,k is recognized as an attractive interconnection networks. Let x, y, and z be three different vertices of A _n,k . Let l be any integer with $d_{A_{n,k}}(\mathbf{x},\mathbf{y}) \le l \le \frac{n!}{(n-k)!}-1-d_{A_{n,k}}(\mathbf{y},\mathbf{z})$ . We shall prove the following existance properties of Hamiltonian path: (1)?for n?k≥3 or (n,k)=(3,1), there exists a Hamiltonian path R(x,y,z;l) from x to z such that d _R(x,y,z;l)(x,y)=l; (2) for n?k=2 and n≥5, there exists a Hamiltonian path R(x,y,z;l) except for the case that x, y, and z are adjacent to each other.     

Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

A k-colouring of a graph G=(V,E) is a mapping c:V→{1,2,…,k} such that c(u)≠c(v) whenever uv is an edge. The reconfiguration graph of the k-colourings of G contains as its vertex set the k-colourings of G, and two colourings are joined by an edge if they differ in colour on just one vertex of G. We introduce a class of k-colourable graphs, which we call k-colour-dense graphs. We show that for each k-colour-dense graph G, the reconfiguration graph of the ?-colourings of G is connected and has diameter O(|V|²), for all ?≥k+1. We show that this graph class contains the k-colourable chordal graphs and that it contains all chordal bipartite graphs when k=2. Moreover, we prove that for each k≥2 there is a k-colourable chordal graph G whose reconfiguration graph of the (k+1)-colourings has diameter Θ(|V|²).     

A sufficient condition for planar graphs to be (3, 1)-choosable

A (k, d)-list assignment L of a graph is a function that assigns to each vertex v a list L(v) of at least k colors satisfying \(|L(x)\cap L(y)|\le d\) for each edge xy. An L-coloring is a vertex coloring \(\pi \) such that \(\pi (v) \in L(v)\) for each vertex v and \(\pi (x) \ne \pi (y)\) for each edge xy. A graph G is (k, d)-choosable if there exists an L-coloring of G for every (k, d)-list assignment L. This concept is known as choosability with separation. In this paper, we will use Thomassen list coloring extension method to prove that planar graphs with neither 6-cycles nor adjacent 4- and 5-cycles are (3, 1)-choosable. This is a strengthening of a result obtained by using Discharging method which says that planar graphs without 5- and 6-cycles are (3, 1)-choosable.     

On total weight choosability of graphs

For a graph G with vertex set V and edge set E, a (k,k′)-total list assignment L of G assigns to each vertex v a set L(v) of k real numbers as permissible weights, and assigns to each edge e a set L(e) of k′ real numbers as permissible weights. If for any (k,k′)-total list assignment L of G, there exists a mapping f:V∪E→? such that f(y)∈L(y) for each y∈V∪E, and for any two adjacent vertices u and v, ∑ _y∈N(u) f(uy)+f(u)≠∑ _x∈N(v) f(vx)+f(v), then G is (k,k′)-total weight choosable. It is conjectured by Wong and Zhu that every graph is (2,2)-total weight choosable, and every graph with no isolated edges is (1,3)-total weight choosable. In this paper, it is proven that a graph G obtained from any loopless graph H by subdividing each edge with at least one vertex is (1,3)-total weight choosable and (2,2)-total weight choosable. It is shown that s-degenerate graphs (with s≥2) are (1,2s)-total weight choosable. Hence planar graphs are (1,10)-total weight choosable, and outerplanar graphs are (1,4)-total weight choosable. We also give a combinatorial proof that wheels are (2,2)-total weight choosable, as well as (1,3)-total weight choosable.